Projective Rank-One Reduction for Terminal Navier–Stokes Saturation Scenarios

Version 2.0 substantially revises and retitles the earlier paper Projective Rank-one Closure for Terminal Navier–Stokes Saturation. The revised work studies terminal rank-one output-coherence mechanisms in a dyadic analysis of the three-dimensional incompressible Navier–Stokes equations and replaces the earlier closure claim with a branch-reduction and routing theorem.The analysis separates comparable high–high interactions from a complementary determining-scale paraproduct strain branch and uses localized output Gram matrices to measure projective coherence of the nonlinear output. The corrected high–high sector is imported as a reduction module: it may yield positive coherence-rank defect, Beltrami depletion, or finite-pattern parabolic damping, while retaining a possible nondepleted near-rank-one projective-output remainder.For the paraproduct branch, multiple active positive strain directions are routed to an imported middle-eigenvalue strain criterion, while mutually orthogonal significant output sectors generate positive coherence-rank defect. When the nonlinear output remains concentrated near one projective direction, an additional conditional moving-frame realization is analyzed through a one-component profile \(U=\phi v\), \(|v|=1\), whose first-order projective motion is represented by \((I-v\otimes v)\nabla v=\nabla v\). Residual near-rank-one output and collapsed-projective-motion branches are retained explicitly rather than treated as closed.Accordingly, Version 2.0 should be read as a terminal branch-reduction and routing result identifying the remaining analytic sectors that require further control, rather than as a complete Navier–Stokes continuation or regularity closure theorem.